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field; subfield; isomorphism; transcendental extension; algebraic extension
In this note we study fields $F$ with the property that the simple transcendental extension $F(u)$ of $F$ is isomorphic to some subfield of $F$ but not isomorphic to $F$. Such a field provides one type of solution of the Schröder-Bernstein problem for fields.
[1] W. T. Gowers: A solution to the Schröder-Bernstein problem for Banach spaces. Bull. London Math. Soc. 28 (1996), 297–304. DOI 10.1112/blms/28.3.297 | MR 1374409 | Zbl 0863.46006
[2] I. Kaplansky: Infinite Abelian Groups. Revised edition, University of Michigan Press, 1969. MR 0233887 | Zbl 0194.04402
[3] J. Kelley: General Topology. D. van Nostrand, New York, 1955. MR 0070144 | Zbl 0066.16604
[4] B. L. van der Waerden: Modern Algebra. Vol. 1. Ungar, New York, 1953.
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